1
vote
1answer
27 views

$H^1(X) = [X,\mathbb{T}]$?

This is a stupid question, but here goes. I have a compact Hausdorff space $X$, and I am talking about $[X,\mathbb{T}]$, the group of homotopy classes of maps $X \to \mathbb{T}$, where $\mathbb{T}$ ...
2
votes
0answers
34 views

Is it possible to call such properties “cocycle conditions”?

Let $\mathcal O = \{\alpha,\beta,\ldots\}$ be a family of objects of general but similar type (numbers, functions,e.t.c.). Let $\{ X_\alpha \mid \alpha \in \mathcal O\}$ be a family of linear spaces ...
2
votes
1answer
96 views

Why functions $f_{\alpha \beta} \colon U_\alpha \cap U_\beta \to Y$ are called cocycles?

Let $X$ be some smooth manifold and $\{U_\alpha\}$ be its open cover. The last month I hear very often that one calls a collection of functions $f_{\alpha \beta} \colon U_\alpha \cap U_\beta \to Y$, ...
7
votes
2answers
227 views

What is the motivation for defining both homogeneous and inhomogeneous cochains?

In my few months of studying group cohomology, I've seen two "standard" complexes that are introduced: We let $X_r$ be the free $\mathbb{Z}[G]$-module on $G^r$ (so, it has as a $\mathbb{Z}[G]$-basis ...
15
votes
2answers
687 views

Meaning of “efface” in “effaceable functor” and “injective effacement”

I'm reading Grothendieck's Tōhoku paper, and I was curious about the reasoning behind the terms "effaceable functor" and "injective effacement". I know that in English, to efface something means ...